24 | January | 2008 | A kind of library

116b- Lecture 6

January 24, 2008

We proved that a set is recursive iff it is $\Delta_1$ -definable.

We showed some elementary properties of the theory $\mathsf{Q}$ , defined end-extensions, and verified that $\mathsf{Q}$ is $\Sigma_1$ -complete.

We also defined what it means for a function, or a set, to be represented in a theory.

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- Answer by Andrés E. Caicedo for Where is the Erdős–Rado theorem stated in Erdős and Rado's Bull AMS paper? December 23, 2014
  This is Theorem 39 in the paper (see Theorem 4.(i) for a user-friendly preview). But the fact that $(2^\kappa)^+\to(\kappa^+)^2_\kappa$ is older (1946) and due to Erdős, see here: Paul Erdős. Some set-theoretical properties of graphs, Univ. Nac. Tucumán. Revista A. 3 (1942), 363-367 MR0009444 (5,151d). (Anyway, it is probably easier to read a more modern pre […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for The Erdős-Turán conjecture or the Erdős' conjecture? June 3, 2013
  One of the best places to track these things down is The mathematical coloring book, by Alexander Soifer, Springer 2009. Chapter 35 is on "Monochromatic arithmetic progressions", and section 35.4, "Paul Erdős’s Favorite Conjecture", is on the problem you ask about. As far as I can tell, the question is sometimes called the Erdős-Turán con […]
  Andrés E. Caicedo
- Relative densities September 12, 2013
  Throughout the question, we only consider primes of the form $3k+1$. A reference for cubic reciprocity is Ireland & Rosen's A Classical Introduction to Modern Number Theory. How can I count the relative density of those $p$ (of the form $3k+1$) such that the equation $2=3x^3$ has no solutions modulo $p$? Really, even pointers on how to say anything […]
  Andrés E. Caicedo
- On the division paradox December 22, 2022
  This question is partly motivated by Timothy Chow's recent question on the division paradox. Say that a set $X$ admits a paradoxical partition if and only if there is an equivalence relation $\sim$ on $X$ such that $|X|
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Relative densities December 22, 2022
  A solution can be obtained as suggested by Keith Conrad in the comments, via Chebotarëv's theorem. Details can be found in $\S3.4$ of Coloring the $n$-Smooth Numbers with $n$ Colors Andrés Eduardo Caicedo, Thomas A. C. Chartier, Péter Pál Pach The Electronic Journal of Combinatorics 28 (1) (2021), #P1.34, 79 pp. DOI: https://doi.org/10.37236/8492 Many t […]
  Andrés E. Caicedo
Math.StackExchange
- Answer by Andrés E. Caicedo for Is it possible for a function to be smooth everywhere, analytic nowhere, yet Taylor series at any point converges in a nonzero radius? January 7, 2014
  No, this is not possible. Dave L. Renfro wrote an excellent historical Essay on nowhere analytic $C^\infty$ functions in two parts (with numerous references). See here: 1 (dated May 9, 2002 6:18 PM), and 2 (dated May 19, 2002 8:29 PM). As indicated in part 1, in Zygmunt Zahorski. Sur l'ensemble des points singuliers d'une fonction d'une variab […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for How to check the real analyticity of a function? December 3, 2013
  This is a difficult question in general. Ideally, to show that $f$ is analytic at the origin, you show that in a suitable neighborhood of $0$, the error of the $n$-th Taylor polynomial approaches $0$ as $n\to\infty$. For example, for $f(x)=\sin(x)$, any derivative of $f(x)$ is one of $\sin(x)$, $\cos(x)$, $-\sin(x)$, or $-\cos(x)$, and the error given by the […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Borel hierarchy doesn't "collapse" before $\omega_1$ October 1, 2013
  To complement Yann's answer: This is a nice problem, the possible length of Borel hierarchies in different spaces or without assuming the axiom of choice. It has been studied in detail by Arnie Miller. See Arnold W. Miller. On the length of Borel hierarchies, Ann. Math. Logic, 16 (3), (1979), 233–267. MR0548475 (80m:04003), Arnold W. Miller. Long Borel […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for consistency strength of PA September 3, 2013
  This is a good question, because a priori $\mathsf{PA}$ lacks the flexibility of $\mathsf{ZFC}$ that allows us to deal with consistency problems semantically (by building models) and, anyway, the obvious model of most subtheories of $\mathsf{PA}$ is just the standard model. The way this is done in the context of $\mathsf{ZFC}$ is using the reflection theorem […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Does ZFC prove a sentence in the language of arithmetic that PA+Con(ZFC) cannot prove? August 31, 2013
  Yes, of course. An example is the statement that all Goodstein sequences terminate. The point is that this sentence is not only independent of $\mathsf{PA}$, but in fact of the theory resulting from adding to $\mathsf{PA}$ all $\Pi^0_1$ statements true in the standard model of arithmetic. Note that $\mathrm{Con}(\mathsf{ZFC})$ is an example of such a $\Pi^0_ […]
  Andrés E. Caicedo
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116b- Lecture 6

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